"""
DaoMath Field Theory (道数场论)
从 ±0 先天均衡构造场的完整理论

场论的 ±0 诠释:
    场 ↔ 源 (对偶)
    粒子 ↔ 激发 (对偶)
    规范对称 ↔ 守恒流 (对偶)
    
理论层级:
    ±0 (先天均衡)
     ↓
    场配置 (Field Configuration)
     ↓
    Lagrangian 密度
     ↓
    运动方程 (Euler-Lagrange)
     ↓
    对称性与守恒律 (Noether)
     ↓
    规范场 (Gauge Fields)
     ↓
    Yang-Mills 理论
     ↓
    量子化 (Path Integral)
"""

from __future__ import annotations
from typing import (
    List, Tuple, Callable, Optional, Dict, Any,
    Union, TypeVar, Generic
)
from dataclasses import dataclass, field
from abc import ABC, abstractmethod
from enum import Enum
import numpy as np
import scipy.linalg as la
from scipy.integrate import odeint, quad
from scipy.optimize import minimize
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

from core.axioms import InnateBalance, DualPair
from structures.algebra import LieAlgebra, LieGroup


# ============================================================================
# 基础结构
# ============================================================================

class SpacetimePoint:
    """时空点 x^μ = (t, x, y, z)"""
    
    def __init__(self, t: float, x: float, y: float, z: float):
        self.coords = np.array([t, x, y, z])
    
    def __getitem__(self, idx: int) -> float:
        return float(self.coords[idx])
    
    def __repr__(self) -> str:
        return f"({self.coords[0]:.3f}, {self.coords[1]:.3f}, " \
               f"{self.coords[2]:.3f}, {self.coords[3]:.3f})"


class LorentzIndex:
    """Lorentz 指标类型"""
    CONTRAVARIANT = "upper"  # 上指标 μ
    COVARIANT = "lower"      # 下指标 μ


# ============================================================================
# Level 1: 场的基本定义
# ============================================================================

class Field(ABC):
    """
    场的抽象基类
    
    场：时空点到某个值空间的映射
    φ: ℝ⁴ → V
    
    基于 ±0 的理解:
    - 场是空间的属性分布
    - 场激发对应粒子
    - 场的对偶是源
    """
    
    @abstractmethod
    def value_at(self, point: np.ndarray) -> Any:
        """在给定时空点的场值"""
        pass
    
    @abstractmethod
    def dimension(self) -> int:
        """场的维数"""
        pass
    
    @abstractmethod
    def derivative(self, point: np.ndarray, mu: int) -> Any:
        """偏导数 ∂_μ φ"""
        pass


class ScalarField(Field):
    """
    标量场 φ(x)
    
    最简单的场：φ: ℝ⁴ → ℝ
    
    实例: Higgs 场
    """
    
    def __init__(self, field_function: Callable[[np.ndarray], float]):
        """
        Args:
            field_function: x^μ → φ(x) 的函数
        """
        self.field_func = field_function
    
    def value_at(self, point: np.ndarray) -> float:
        """场值 φ(x)"""
        return float(self.field_func(point))
    
    def dimension(self) -> int:
        return 1
    
    def derivative(self, point: np.ndarray, mu: int, delta: float = 1e-6) -> float:
        """
        偏导数 ∂_μ φ (数值微分)
        """
        point_plus = point.copy()
        point_plus[mu] += delta
        
        phi_plus = self.value_at(point_plus)
        phi = self.value_at(point)
        
        return (phi_plus - phi) / delta
    
    def gradient(self, point: np.ndarray) -> np.ndarray:
        """四维梯度 ∂_μ φ"""
        return np.array([self.derivative(point, mu) for mu in range(4)])
    
    def laplacian(self, point: np.ndarray, delta: float = 1e-6) -> float:
        """
        d'Alembertian 算符
        □φ = ∂_μ ∂^μ φ = -∂²_t φ + ∇²φ
        """
        # 时间二阶导
        t_second = (self.value_at(point + np.array([delta, 0, 0, 0])) -
                   2 * self.value_at(point) +
                   self.value_at(point - np.array([delta, 0, 0, 0]))) / delta**2
        
        # 空间 Laplacian
        spatial_laplacian = 0.0
        for i in range(1, 4):
            direction = np.zeros(4)
            direction[i] = delta
            spatial_laplacian += (self.value_at(point + direction) -
                                 2 * self.value_at(point) +
                                 self.value_at(point - direction)) / delta**2
        
        return -t_second + spatial_laplacian
    
    @staticmethod
    def from_innate_balance(amplitude: float = 1.0) -> ScalarField:
        """
        从 ±0 构造标量场
        
        φ(x) = A cos(k·x - ωt)
        体现正负振荡
        """
        k = np.array([0, 1, 0, 0])  # 波矢
        omega = 1.0  # 频率
        
        def field_func(x: np.ndarray) -> float:
            return amplitude * np.cos(np.dot(k, x) - omega * x[0])
        
        return ScalarField(field_func)


class VectorField(Field):
    """
    矢量场 A^μ(x)
    
    四矢量场：A: ℝ⁴ → ℝ⁴
    
    实例: 电磁场 A^μ = (φ, A)
    """
    
    def __init__(self, field_function: Callable[[np.ndarray], np.ndarray]):
        """
        Args:
            field_function: x^μ → A^μ(x)
        """
        self.field_func = field_function
    
    def value_at(self, point: np.ndarray) -> np.ndarray:
        """场值 A^μ(x)"""
        return self.field_func(point)
    
    def dimension(self) -> int:
        return 4
    
    def derivative(self, point: np.ndarray, mu: int, 
                  delta: float = 1e-6) -> np.ndarray:
        """
        偏导数 ∂_μ A^ν
        """
        point_plus = point.copy()
        point_plus[mu] += delta
        
        A_plus = self.value_at(point_plus)
        A = self.value_at(point)
        
        return (A_plus - A) / delta
    
    def field_strength(self, point: np.ndarray) -> np.ndarray:
        """
        场强张量 F_μν = ∂_μ A_ν - ∂_ν A_μ
        
        对于电磁场：F^μν = (E, B)
        """
        F = np.zeros((4, 4))
        
        for mu in range(4):
            for nu in range(4):
                dA_nu = self.derivative(point, mu)
                dA_mu = self.derivative(point, nu)
                
                F[mu, nu] = dA_nu[nu] - dA_mu[mu]
        
        return F
    
    def divergence(self, point: np.ndarray) -> float:
        """
        四维散度 ∂_μ A^μ
        """
        div = 0.0
        for mu in range(4):
            div += self.derivative(point, mu)[mu]
        return div
    
    @staticmethod
    def electromagnetic_potential(
        electric_field: np.ndarray,
        magnetic_field: np.ndarray
    ) -> VectorField:
        """
        从 E, B 场构造四势 A^μ
        
        简化：静态场
        """
        def field_func(x: np.ndarray) -> np.ndarray:
            # φ = -∫E·dx (简化)
            phi = -np.dot(electric_field, x[1:])
            
            # A = ½(B × r) (简化)
            A = 0.5 * np.cross(magnetic_field, x[1:])
            
            return np.concatenate([[phi], A])
        
        return VectorField(field_func)


class TensorField(Field):
    """
    张量场 T^μν(x)
    
    二阶张量场
    
    实例: 能量-动量张量 T^μν
    """
    
    def __init__(self, field_function: Callable[[np.ndarray], np.ndarray]):
        """
        Args:
            field_function: x^μ → T^μν(x) (4×4 矩阵)
        """
        self.field_func = field_function
    
    def value_at(self, point: np.ndarray) -> np.ndarray:
        """张量值 T^μν(x)"""
        return self.field_func(point)
    
    def dimension(self) -> int:
        return 16  # 4×4 张量
    
    def derivative(self, point: np.ndarray, mu: int,
                  delta: float = 1e-6) -> np.ndarray:
        """协变导数（简化为偏导）"""
        point_plus = point.copy()
        point_plus[mu] += delta
        
        T_plus = self.value_at(point_plus)
        T = self.value_at(point)
        
        return (T_plus - T) / delta
    
    def trace(self, point: np.ndarray) -> float:
        """迹 T = T^μ_μ"""
        T = self.value_at(point)
        return float(np.trace(T))
    
    def is_symmetric(self, point: np.ndarray, tol: float = 1e-10) -> bool:
        """判断对称性 T^μν = T^νμ"""
        T = self.value_at(point)
        return np.allclose(T, T.T, atol=tol)


# ============================================================================
# Level 2: Lagrangian 与作用量
# ============================================================================

class LagrangianDensity:
    """
    Lagrangian 密度 ℒ(φ, ∂_μφ)
    
    作用量: S = ∫ d⁴x ℒ
    
    基于 ±0 的理解:
    - 动能项（正）vs 势能项（负）的均衡
    - 极值原理：δS = 0
    """
    
    def __init__(self, 
                 lagrangian_func: Callable[[np.ndarray, np.ndarray], float]):
        """
        Args:
            lagrangian_func: (φ, ∂_μφ) → ℒ
        """
        self.lagrangian_func = lagrangian_func
    
    def value(self, field_value: np.ndarray, 
             field_derivative: np.ndarray) -> float:
        """ℒ(φ, ∂_μφ)"""
        return self.lagrangian_func(field_value, field_derivative)
    
    def euler_lagrange_equation(self,
                                field: ScalarField,
                                point: np.ndarray) -> float:
        """
        Euler-Lagrange 方程
        
        ∂ℒ/∂φ - ∂_μ(∂ℒ/∂(∂_μφ)) = 0
        """
        phi = field.value_at(point)
        dphi = field.gradient(point)
        
        # ∂ℒ/∂φ
        delta_phi = 1e-6
        dL_dphi = (self.value(phi + delta_phi, dphi) - 
                  self.value(phi, dphi)) / delta_phi
        
        # ∂_μ(∂ℒ/∂(∂_μφ))
        term2 = 0.0
        for mu in range(4):
            # ∂ℒ/∂(∂_μφ)
            dphi_plus = dphi.copy()
            dphi_plus[mu] += delta_phi
            dL_ddphi = (self.value(phi, dphi_plus) - 
                       self.value(phi, dphi)) / delta_phi
            
            # 对其求导
            point_plus = point.copy()
            point_plus[mu] += delta_phi
            dphi_at_plus = field.gradient(point_plus)
            
            dL_ddphi_plus = (self.value(phi, dphi_at_plus) - 
                            self.value(phi, dphi)) / delta_phi
            
            term2 += (dL_ddphi_plus - dL_ddphi) / delta_phi
        
        return dL_dphi - term2
    
    @staticmethod
    def klein_gordon(mass: float) -> LagrangianDensity:
        """
        Klein-Gordon 场（自由标量场）
        
        ℒ = ½(∂_μφ ∂^μφ) - ½m²φ²
        
        体现动能-势能均衡
        """
        def lagrangian(phi: float, dphi: np.ndarray) -> float:
            # ∂_μφ ∂^μφ = -(∂_tφ)² + (∇φ)²
            kinetic = -dphi[0]**2 + np.sum(dphi[1:]**2)
            potential = mass**2 * phi**2
            
            return 0.5 * (kinetic - potential)
        
        return LagrangianDensity(lagrangian)
    
    @staticmethod
    def phi_fourth(mass: float, lambda_coupling: float) -> LagrangianDensity:
        """
        φ⁴ 理论（自相互作用标量场）
        
        ℒ = ½(∂_μφ ∂^μφ) - ½m²φ² - (λ/4!)φ⁴
        """
        def lagrangian(phi: float, dphi: np.ndarray) -> float:
            kinetic = -dphi[0]**2 + np.sum(dphi[1:]**2)
            mass_term = mass**2 * phi**2
            interaction = (lambda_coupling / 24) * phi**4
            
            return 0.5 * (kinetic - mass_term) - interaction
        
        return LagrangianDensity(lagrangian)
    
    @staticmethod
    def electromagnetic() -> LagrangianDensity:
        """
        电磁场
        
        ℒ = -¼ F_μν F^μν
        
        其中 F_μν = ∂_μ A_ν - ∂_ν A_μ
        """
        def lagrangian(A: np.ndarray, dA: np.ndarray) -> float:
            # 构造场强张量 F_μν
            F = np.zeros((4, 4))
            for mu in range(4):
                for nu in range(4):
                    F[mu, nu] = dA[mu, nu] - dA[nu, mu]
            
            # F_μν F^μν (Minkowski 度规)
            eta = np.diag([-1, 1, 1, 1])
            F_squared = 0.0
            for mu in range(4):
                for nu in range(4):
                    for rho in range(4):
                        for sigma in range(4):
                            F_squared += (eta[mu, rho] * eta[nu, sigma] * 
                                        F[mu, nu] * F[rho, sigma])
            
            return -0.25 * F_squared
        
        return LagrangianDensity(lagrangian)


class Action:
    """
    作用量 S[φ] = ∫ d⁴x ℒ(φ, ∂_μφ)
    
    经典场通过极值原理确定
    """
    
    def __init__(self, lagrangian: LagrangianDensity):
        self.lagrangian = lagrangian
    
    def evaluate(self,
                field: ScalarField,
                spacetime_region: Tuple[np.ndarray, np.ndarray],
                num_points: int = 20) -> float:
        """
        计算作用量（数值积分）
        
        Args:
            spacetime_region: (x_min, x_max) 时空区域
            num_points: 每个维度的采样点数
        """
        x_min, x_max = spacetime_region
        
        # 生成网格
        t_grid = np.linspace(x_min[0], x_max[0], num_points)
        x_grid = np.linspace(x_min[1], x_max[1], num_points)
        y_grid = np.linspace(x_min[2], x_max[2], num_points)
        z_grid = np.linspace(x_min[3], x_max[3], num_points)
        
        action_value = 0.0
        dt = (x_max[0] - x_min[0]) / num_points
        dx = (x_max[1] - x_min[1]) / num_points
        dy = (x_max[2] - x_min[2]) / num_points
        dz = (x_max[3] - x_min[3]) / num_points
        d4x = dt * dx * dy * dz
        
        for t in t_grid:
            for x in x_grid:
                for y in y_grid:
                    for z in z_grid:
                        point = np.array([t, x, y, z])
                        phi = field.value_at(point)
                        dphi = field.gradient(point)
                        
                        L = self.lagrangian.value(phi, dphi)
                        action_value += L * d4x
        
        return action_value
    
    def variational_derivative(self,
                              field: ScalarField,
                              point: np.ndarray) -> float:
        """
        变分导数 δS/δφ(x)
        
        等价于 Euler-Lagrange 方程
        """
        return self.lagrangian.euler_lagrange_equation(field, point)


# ============================================================================
# Level 3: 对称性与守恒律 (Noether 定理)
# ============================================================================

class Symmetry(ABC):
    """
    对称性抽象基类
    
    对称性 → 守恒律 (Noether 定理)
    """
    
    @abstractmethod
    def transform(self, field: Field, epsilon: float) -> Field:
        """对称变换"""
        pass
    
    @abstractmethod
    def conserved_current(self, field: Field, point: np.ndarray) -> np.ndarray:
        """守恒流 j^μ"""
        pass


class TranslationSymmetry(Symmetry):
    """
    平移对称性
    
    x^μ → x^μ + a^μ
    
    守恒律：能量-动量守恒
    """
    
    def __init__(self, direction: np.ndarray):
        """
        Args:
            direction: 平移方向 a^μ
        """
        self.direction = direction / np.linalg.norm(direction)
    
    def transform(self, field: ScalarField, epsilon: float) -> ScalarField:
        """场的平移变换"""
        translation = epsilon * self.direction
        
        def transformed_field(x: np.ndarray) -> float:
            return field.value_at(x - translation)
        
        return ScalarField(transformed_field)
    
    def conserved_current(self, 
                         field: ScalarField,
                         lagrangian: LagrangianDensity,
                         point: np.ndarray) -> np.ndarray:
        """
        能量-动量张量 T^μν
        
        T^μν = ∂ℒ/∂(∂_μφ) ∂^νφ - η^μν ℒ
        """
        phi = field.value_at(point)
        dphi = field.gradient(point)
        
        T = np.zeros((4, 4))
        eta = np.diag([-1, 1, 1, 1])
        
        L = lagrangian.value(phi, dphi)
        
        for mu in range(4):
            for nu in range(4):
                # ∂ℒ/∂(∂_μφ)
                delta = 1e-6
                dphi_plus = dphi.copy()
                dphi_plus[mu] += delta
                dL_ddphi = (lagrangian.value(phi, dphi_plus) - L) / delta
                
                T[mu, nu] = dL_ddphi * dphi[nu] - eta[mu, nu] * L
        
        return T


class RotationSymmetry(Symmetry):
    """
    旋转对称性
    
    守恒律：角动量守恒
    """
    
    def __init__(self, axis: np.ndarray):
        """
        Args:
            axis: 旋转轴（3D）
        """
        self.axis = axis / np.linalg.norm(axis)
    
    def transform(self, field: ScalarField, epsilon: float) -> ScalarField:
        """场的旋转变换（仅空间坐标）"""
        def rotation_matrix(angle: float) -> np.ndarray:
            # Rodrigues 公式
            K = np.array([
                [0, -self.axis[2], self.axis[1]],
                [self.axis[2], 0, -self.axis[0]],
                [-self.axis[1], self.axis[0], 0]
            ])
            return (np.eye(3) + np.sin(angle) * K + 
                   (1 - np.cos(angle)) * K @ K)
        
        R = rotation_matrix(epsilon)
        
        def transformed_field(x: np.ndarray) -> float:
            x_rotated = np.concatenate([
                [x[0]],  # 时间不变
                R.T @ x[1:]  # 空间旋转
            ])
            return field.value_at(x_rotated)
        
        return ScalarField(transformed_field)
    
    def conserved_current(self,
                         field: ScalarField,
                         lagrangian: LagrangianDensity,
                         point: np.ndarray) -> np.ndarray:
        """
        角动量密度 M^μνρ = x^ν T^μρ - x^ρ T^μν
        """
        # 获取能量-动量张量
        translation = TranslationSymmetry(np.array([1, 0, 0, 0]))
        T = translation.conserved_current(field, lagrangian, point)
        
        # 构造角动量张量
        M = np.zeros((4, 4, 4))
        for mu in range(4):
            for nu in range(4):
                for rho in range(4):
                    M[mu, nu, rho] = point[nu] * T[mu, rho] - point[rho] * T[mu, nu]
        
        return M


class GaugeSymmetry(Symmetry):
    """
    规范对称性（最重要！）
    
    局域对称性：变换依赖时空点
    
    U(1): 电磁场
    SU(2): 弱相互作用
    SU(3): 强相互作用（QCD）
    """
    
    def __init__(self, gauge_group: str = 'U1'):
        """
        Args:
            gauge_group: 'U1', 'SU2', 'SU3'
        """
        self.gauge_group = gauge_group
    
    def transform(self, field: VectorField, gauge_param: Callable) -> VectorField:
        """
        规范变换
        
        U(1): A_μ → A_μ + ∂_μ Λ
        """
        if self.gauge_group == 'U1':
            def transformed_field(x: np.ndarray) -> np.ndarray:
                A = field.value_at(x)
                
                # 计算 ∂_μ Λ
                delta = 1e-6
                dLambda = np.zeros(4)
                for mu in range(4):
                    x_plus = x.copy()
                    x_plus[mu] += delta
                    dLambda[mu] = (gauge_param(x_plus) - gauge_param(x)) / delta
                
                return A + dLambda
            
            return VectorField(transformed_field)
        
        raise NotImplementedError(f"Gauge group {self.gauge_group} not implemented")
    
    def conserved_current(self,
                         field: VectorField,
                         point: np.ndarray) -> np.ndarray:
        """
        守恒流（电荷流）
        
        j^μ = ∂_ν F^μν
        """
        F = field.field_strength(point)
        
        j = np.zeros(4)
        for mu in range(4):
            for nu in range(4):
                # ∂_ν F^μν (数值微分)
                delta = 1e-6
                point_plus = point.copy()
                point_plus[nu] += delta
                F_plus = field.field_strength(point_plus)
                
                j[mu] += (F_plus[mu, nu] - F[mu, nu]) / delta
        
        return j


class NoetherTheorem:
    """
    Noether 定理
    
    连续对称性 ↔ 守恒律
    
    基于 ±0 的理解:
    - 对称变换：保持作用量不变（均衡）
    - 守恒流：对称性的具体体现
    """
    
    @staticmethod
    def verify_conservation(
        symmetry: Symmetry,
        field: ScalarField,
        lagrangian: LagrangianDensity,
        spacetime_region: Tuple[np.ndarray, np.ndarray]
    ) -> bool:
        """
        验证守恒律
        
        ∂_μ j^μ = 0
        """
        x_min, x_max = spacetime_region
        test_point = (x_min + x_max) / 2
        
        j = symmetry.conserved_current(field, lagrangian, test_point)
        
        # 计算散度
        div_j = 0.0
        delta = 1e-6
        
        for mu in range(4):
            point_plus = test_point.copy()
            point_plus[mu] += delta
            j_plus = symmetry.conserved_current(field, lagrangian, point_plus)
            
            div_j += (j_plus[mu] - j[mu]) / delta
        
        return abs(div_j) < 1e-4


# ============================================================================
# Level 4: 规范场论 (Gauge Field Theory)
# ============================================================================

class GaugeField:
    """
    规范场 A_μ
    
    作为规范对称性的"补偿场"
    
    协变导数: D_μ = ∂_μ - ig A_μ
    """
    
    def __init__(self, 
                 vector_field: VectorField,
                 gauge_group: str = 'U1',
                 coupling: float = 1.0):
        """
        Args:
            vector_field: 规范势 A_μ
            gauge_group: 规范群
            coupling: 耦合常数 g
        """
        self.A = vector_field
        self.gauge_group = gauge_group
        self.g = coupling
    
    def covariant_derivative(self,
                            scalar_field: ScalarField,
                            point: np.ndarray,
                            mu: int) -> complex:
        """
        协变导数
        
        D_μ φ = ∂_μ φ - ig A_μ φ
        """
        partial_derivative = scalar_field.derivative(point, mu)
        A_mu = self.A.value_at(point)[mu]
        phi = scalar_field.value_at(point)
        
        return partial_derivative - 1j * self.g * A_mu * phi
    
    def field_strength_tensor(self, point: np.ndarray) -> np.ndarray:
        """
        场强张量
        
        F_μν = ∂_μ A_ν - ∂_ν A_μ + ig[A_μ, A_ν]
        
        对于 U(1): 对易子为零
        """
        F = self.A.field_strength(point)
        
        if self.gauge_group == 'U1':
            return F  # U(1) 是 Abelian
        
        # 非 Abelian: 添加对易子项
        # [A_μ, A_ν] (需要李代数结构)
        return F
    
    def gauge_transform(self, gauge_function: Callable[[np.ndarray], float]):
        """
        规范变换
        
        A_μ → A_μ + (1/g) ∂_μ Λ
        """
        def transformed_potential(x: np.ndarray) -> np.ndarray:
            A = self.A.value_at(x)
            
            # 计算 ∂_μ Λ
            delta = 1e-6
            dLambda = np.zeros(4)
            for mu in range(4):
                x_plus = x.copy()
                x_plus[mu] += delta
                dLambda[mu] = (gauge_function(x_plus) - gauge_function(x)) / delta
            
            return A + dLambda / self.g
        
        return GaugeField(VectorField(transformed_potential), 
                         self.gauge_group, self.g)


class YangMillsField:
    """
    Yang-Mills 场（非 Abelian 规范场）
    
    SU(N) 规范理论的推广
    
    F^a_μν = ∂_μ A^a_ν - ∂_ν A^a_μ + g f^abc A^b_μ A^c_ν
    
    其中 f^abc 是李代数结构常数
    """
    
    def __init__(self,
                 gauge_potentials: List[VectorField],
                 lie_algebra: LieAlgebra,
                 coupling: float = 1.0):
        """
        Args:
            gauge_potentials: A^a_μ (每个生成元一个)
            lie_algebra: 李代数（定义结构常数）
            coupling: 耦合常数 g
        """
        self.A_fields = gauge_potentials
        self.lie_algebra = lie_algebra
        self.g = coupling
        self.dim = len(gauge_potentials)
    
    def field_strength(self, point: np.ndarray) -> np.ndarray:
        """
        非 Abelian 场强
        
        F^a_μν = ∂_μ A^a_ν - ∂_ν A^a_μ + g f^abc A^b_μ A^c_ν
        """
        F_full = np.zeros((self.dim, 4, 4))
        
        for a in range(self.dim):
            # Abelian 部分
            F_abelian = self.A_fields[a].field_strength(point)
            
            # 非 Abelian 贡献
            F_nonabelian = np.zeros((4, 4))
            
            for b in range(self.dim):
                for c in range(self.dim):
                    # 结构常数 f^abc
                    f_abc = self.lie_algebra.structure_constants(a, b, c)
                    
                    A_b = self.A_fields[b].value_at(point)
                    A_c = self.A_fields[c].value_at(point)
                    
                    # f^abc A^b_μ A^c_ν
                    for mu in range(4):
                        for nu in range(4):
                            F_nonabelian[mu, nu] += f_abc * A_b[mu] * A_c[nu]
            
            F_full[a] = F_abelian + self.g * F_nonabelian
        
        return F_full
    
    def yang_mills_lagrangian(self, point: np.ndarray) -> float:
        """
        Yang-Mills Lagrangian
        
        ℒ_YM = -¼ Tr(F_μν F^μν) = -¼ F^a_μν F^aμν
        """
        F = self.field_strength(point)
        
        # 求和 F^a_μν F^aμν
        eta = np.diag([-1, 1, 1, 1])
        L = 0.0
        
        for a in range(self.dim):
            for mu in range(4):
                for nu in range(4):
                    for rho in range(4):
                        for sigma in range(4):
                            L += (eta[mu, rho] * eta[nu, sigma] * 
                                 F[a, mu, nu] * F[a, rho, sigma])
        
        return -0.25 * L
    
    def instanton_number(self, spacetime_region: Tuple[np.ndarray, np.ndarray]) -> int:
        """
        Instanton 数（拓扑荷）
        
        Q = (1/16π²) ∫ d⁴x ϵ^μνρσ Tr(F_μν F_ρσ)
        """
        # 简化实现：返回拓扑数
        return 0  # 需要完整的拓扑分析


class QCDField(YangMillsField):
    """
    量子色动力学 (Quantum Chromodynamics)
    
    SU(3) Yang-Mills 理论 + 夸克场
    """
    
    def __init__(self, 
                 gauge_potentials: List[VectorField],
                 coupling: float = 1.0):
        """
        Args:
            gauge_potentials: 8 个胶子场 (SU(3) 有 8 个生成元)
            coupling: 强耦合常数 g_s
        """
        # 构造 SU(3) 李代数
        su3_algebra = self._build_su3_algebra()
        
        super().__init__(gauge_potentials, su3_algebra, coupling)
    
    def _build_su3_algebra(self) -> LieAlgebra:
        """构造 SU(3) 李代数（Gell-Mann 矩阵）"""
        # 8 个 Gell-Mann 矩阵
        lambda_matrices = self._gell_mann_matrices()
        
        # 计算结构常数 [λ_a, λ_b] = 2i f_abc λ_c
        return LieAlgebra(generators=lambda_matrices)
    
    def _gell_mann_matrices(self) -> List[np.ndarray]:
        """Gell-Mann 矩阵（SU(3) 生成元）"""
        lambda_1 = np.array([[0, 1, 0], [1, 0, 0], [0, 0, 0]])
        lambda_2 = np.array([[0, -1j, 0], [1j, 0, 0], [0, 0, 0]])
        lambda_3 = np.array([[1, 0, 0], [0, -1, 0], [0, 0, 0]])
        lambda_4 = np.array([[0, 0, 1], [0, 0, 0], [1, 0, 0]])
        lambda_5 = np.array([[0, 0, -1j], [0, 0, 0], [1j, 0, 0]])
        lambda_6 = np.array([[0, 0, 0], [0, 0, 1], [0, 1, 0]])
        lambda_7 = np.array([[0, 0, 0], [0, 0, -1j], [0, 1j, 0]])
        lambda_8 = np.array([[1, 0, 0], [0, 1, 0], [0, 0, -2]]) / np.sqrt(3)
        
        return [lambda_1, lambda_2, lambda_3, lambda_4,
                lambda_5, lambda_6, lambda_7, lambda_8]
    
    def running_coupling(self, energy_scale: float) -> float:
        """
        运行耦合常数（渐近自由）
        
        g_s(μ) = g_s(μ_0) / √(1 + (b_0 g_s²(μ_0)/(4π)) ln(μ/μ_0))
        
        其中 b_0 = 11 - 2n_f/3 (n_f = 夸克味数)
        """
        mu_0 = 1.0  # 参考能标
        g_0 = self.g
        n_f = 6  # 6 个夸克味
        
        b_0 = (11 - 2 * n_f / 3) / (4 * np.pi)
        
        denominator = 1 + b_0 * g_0**2 * np.log(energy_scale / mu_0)
        
        if denominator <= 0:
            return float('inf')  # 非微扰区域
        
        return g_0 / np.sqrt(denominator)
    
    def confinement_scale(self) -> float:
        """
        禁闭能标 Λ_QCD
        
        强相互作用变得非微扰的能标
        """
        return 0.2  # GeV (典型值)


# ============================================================================
# Level 5: 标准模型的规范结构
# ============================================================================

class StandardModelGauge:
    """
    标准模型的规范群
    
    SU(3)_C × SU(2)_L × U(1)_Y
    
    - SU(3)_C: 色规范群（QCD）
    - SU(2)_L: 弱同位旋
    - U(1)_Y: 超荷
    """
    
    def __init__(self):
        self.gauge_groups = {
            'SU3_color': 'SU3',
            'SU2_weak': 'SU2',
            'U1_hypercharge': 'U1'
        }
        
        # 耦合常数
        self.g_s = 1.0   # 强耦合
        self.g = 0.65    # 弱耦合
        self.g_prime = 0.35  # 超荷耦合
    
    def electroweak_mixing_angle(self) -> float:
        """
        Weinberg 角（电弱混合角）
        
        tan θ_W = g'/g
        """
        return np.arctan(self.g_prime / self.g)
    
    def electromagnetic_coupling(self) -> float:
        """
        电磁耦合常数
        
        e = g sin θ_W = g' cos θ_W
        """
        theta_W = self.electroweak_mixing_angle()
        return self.g * np.sin(theta_W)
    
    def higgs_mechanism(self, vacuum_expectation: float = 246.0) -> Dict[str, float]:
        """
        Higgs 机制（自发对称破缺）
        
        SU(2)_L × U(1)_Y → U(1)_EM
        
        Args:
            vacuum_expectation: Higgs VEV (GeV)
        
        Returns:
            粒子质量字典
        """
        v = vacuum_expectation
        
        # W, Z 玻色子质量
        m_W = 0.5 * self.g * v
        m_Z = 0.5 * np.sqrt(self.g**2 + self.g_prime**2) * v
        
        # 费米子质量（通过 Yukawa 耦合）
        # 简化：仅给出数量级
        yukawa_top = 1.0  # 顶夸克 Yukawa 最大
        m_top = yukawa_top * v / np.sqrt(2)
        
        return {
            'W_boson': m_W,
            'Z_boson': m_Z,
            'top_quark': m_top,
            'photon': 0.0,  # 光子无质量
            'gluon': 0.0    # 胶子无质量
        }
    
    def unification_scale(self) -> float:
        """
        大统一能标（GUT scale）
        
        三个耦合常数在此能标汇合
        """
        # 简化估计
        return 1e16  # GeV


# ============================================================================
# Level 6: 量子化与路径积分
# ============================================================================

class PathIntegral:
    """
    路径积分量子化
    
    Z = ∫ 𝒟φ e^(iS[φ]/ℏ)
    
    基于 ±0 的理解:
    - 所有路径的叠加（正负相位干涉）
    - 经典路径：相位极值（均衡点）
    """
    
    def __init__(self, action: Action):
        self.action = action
    
    def partition_function(self,
                          field_configs: List[ScalarField],
                          spacetime_region: Tuple[np.ndarray, np.ndarray],
                          hbar: float = 1.0) -> complex:
        """
        配分函数（简化计算）
        
        Z = Σ_i e^(iS_i/ℏ)
        """
        Z = 0.0 + 0.0j
        
        for field in field_configs:
            S = self.action.evaluate(field, spacetime_region)
            Z += np.exp(1j * S / hbar)
        
        return Z
    
    def green_function(self,
                      point1: np.ndarray,
                      point2: np.ndarray) -> complex:
        """
        Green 函数（传播子）
        
        G(x, y) = ⟨0|T φ(x)φ(y)|0⟩
        """
        # 简化：自由场传播子
        dx = point2 - point1
        r = np.linalg.norm(dx)
        
        if r < 1e-10:
            return complex(float('inf'), 0)
        
        # Feynman 传播子（简化形式）
        return complex(1 / (4 * np.pi * r), 0)
    
    def vacuum_expectation_value(self,
                                operator: Callable[[ScalarField], float],
                                field_configs: List[ScalarField]) -> float:
        """
        真空期望值
        
        ⟨O⟩ = ∫ 𝒟φ O[φ] e^(iS[φ]/ℏ) / Z
        """
        numerator = 0.0
        denominator = 0.0
        
        for field in field_configs:
            S = self.action.evaluate(field, (np.zeros(4), np.ones(4)))
            weight = np.exp(1j * S)
            
            numerator += operator(field) * weight
            denominator += weight
        
        return float(np.real(numerator / denominator))


# ============================================================================
# 综合演示系统
# ============================================================================

class FieldTheoryDemo:
    """场论综合演示"""
    
    @staticmethod
    def demonstrate_scalar_field():
        """演示标量场"""
        print("\n" + "=" * 70)
        print("标量场演示")
        print("=" * 70)
        
        # 创建波动解
        phi = ScalarField.from_innate_balance(amplitude=1.0)
        
        print("\n场配置: φ(x) = cos(k·x - ωt)")
        
        # 计算场值和导数
        x = np.array([0, 0, 0, 0])
        phi_val = phi.value_at(x)
        grad_phi = phi.gradient(x)
        
        print(f"在原点:")
        print(f"  φ(0) = {phi_val:.4f}")
        print(f"  ∂_μ φ = {grad_phi}")
        
        # d'Alembertian
        box_phi = phi.laplacian(x)
        print(f"  □φ = {box_phi:.4f}")
    
    @staticmethod
    def demonstrate_lagrangian():
        """演示 Lagrangian 与运动方程"""
        print("\n" + "=" * 70)
        print("Klein-Gordon 场")
        print("=" * 70)
        
        mass = 1.0
        L = LagrangianDensity.klein_gordon(mass)
        
        print(f"\nLagrangian: ℒ = ½(∂_μφ∂^μφ) - ½m²φ²")
        print(f"质量 m = {mass}")
        
        # 创建场
        phi = ScalarField.from_innate_balance()
        
        # 验证运动方程
        x = np.array([0, 1, 0, 0])
        EL = L.euler_lagrange_equation(phi, x)
        
        print(f"\nEuler-Lagrange 方程:")
        print(f"  (□ + m²)φ = {EL:.6f}")
        print(f"  (应接近 0)")
    
    @staticmethod
    def demonstrate_noether_theorem():
        """演示 Noether 定理"""
        print("\n" + "=" * 70)
        print("Noether 定理：对称性与守恒律")
        print("=" * 70)
        
        # 场与 Lagrangian
        phi = ScalarField.from_innate_balance()
        L = LagrangianDensity.klein_gordon(1.0)
        
        # 时间平移对称性 → 能量守恒
        time_translation = TranslationSymmetry(np.array([1, 0, 0, 0]))
        
        print("\n时间平移对称性:")
        x = np.array([0, 0, 0, 0])
        T = time_translation.conserved_current(phi, L, x)
        
        print(f"能量-动量张量 T^μν:")
        print(f"  T^00 (能量密度) = {T[0, 0]:.4f}")
        print(f"  T^0i (动量密度) = {T[0, 1:4]}")
        
        # 验证守恒
        region = (np.array([0, 0, 0, 0]), np.array([1, 1, 1, 1]))
        conserved = NoetherTheorem.verify_conservation(
            time_translation, phi, L, region
        )
        print(f"\n守恒律验证: {conserved}")
    
    @staticmethod
    def demonstrate_gauge_theory():
        """演示规范场论"""
        print("\n" + "=" * 70)
        print("U(1) 规范场论（电磁场）")
        print("=" * 70)
        
        # 创建电磁势
        E = np.array([0, 0, 1])  # 电场
        B = np.array([0, 0, 0])  # 磁场
        
        A = VectorField.electromagnetic_potential(E, B)
        gauge_field = GaugeField(A, 'U1', coupling=1.0)
        
        print("\n电磁场配置:")
        x = np.array([0, 0, 0, 0])
        A_val = A.value_at(x)
        print(f"  四势 A^μ = {A_val}")
        
        # 场强张量
        F = gauge_field.field_strength_tensor(x)
        print(f"\n场强张量 F^μν:")
        print(f"  F^01 (Ex) = {F[0, 1]:.4f}")
        print(f"  F^02 (Ey) = {F[0, 2]:.4f}")
        print(f"  F^03 (Ez) = {F[0, 3]:.4f}")
        
        # 规范变换
        def gauge_func(x):
            return x[0] * x[1]  # 任意规范函数
        
        A_transformed = gauge_field.gauge_transform(gauge_func)
        print("\n规范变换后:")
        A_new = A_transformed.A.value_at(x)
        print(f"  A'^μ = {A_new}")
    
    @staticmethod
    def demonstrate_yang_mills():
        """演示 Yang-Mills 场"""
        print("\n" + "=" * 70)
        print("Yang-Mills 场（非 Abelian）")
        print("=" * 70)
        
        # 创建 SU(2) Yang-Mills
        # 简化：使用 3 个场（对应 3 个生成元）
        def A1_func(x): return np.array([0, x[1], 0, 0])
        def A2_func(x): return np.array([0, 0, x[2], 0])
        def A3_func(x): return np.array([0, 0, 0, x[3]])
        
        A_fields = [
            VectorField(A1_func),
            VectorField(A2_func),
            VectorField(A3_func)
        ]
        
        # 构造 SU(2) 李代数
        class SimpleLieAlgebra(LieAlgebra):
            def structure_constants(self, a, b, c):
                # SU(2): ϵ_abc
                levi_civita = {
                    (0, 1, 2): 1, (1, 2, 0): 1, (2, 0, 1): 1,
                    (1, 0, 2): -1, (2, 1, 0): -1, (0, 2, 1): -1
                }
                return levi_civita.get((a, b, c), 0)
        
        lie_alg = SimpleLieAlgebra(generators=[])
        
        ym_field = YangMillsField(A_fields, lie_alg, coupling=1.0)
        
        print("\nSU(2) Yang-Mills 场")
        x = np.array([0, 1, 1, 1])
        
        # Yang-Mills Lagrangian
        L_YM = ym_field.yang_mills_lagrangian(x)
        print(f"Lagrangian ℒ_YM = {L_YM:.4f}")
        
        print("\n非 Abelian 特性:")
        print("  场强包含自相互作用项 g f^abc A^b_μ A^c_ν")
    
    @staticmethod
    def demonstrate_qcd():
        """演示 QCD"""
        print("\n" + "=" * 70)
        print("量子色动力学 (QCD)")
        print("=" * 70)
        
        # 创建 8 个胶子场（简化）
        gluon_fields = []
        for i in range(8):
            def field_func(x, idx=i):
                result = np.zeros(4)
                result[idx % 4] = x[(idx + 1) % 4]
                return result
            gluon_fields.append(VectorField(field_func))
        
        qcd = QCDField(gluon_fields, coupling=1.0)
        
        print("\nSU(3) 规范群:")
        print("  8 个胶子 (对应 8 个 Gell-Mann 矩阵)")
        
        # 渐近自由
        print("\n运行耦合常数 (渐近自由):")
        energy_scales = [0.1, 1, 10, 100]  # GeV
        for E in energy_scales:
            g_s = qcd.running_coupling(E)
            print(f"  E = {E:6.1f} GeV: g_s = {g_s:.4f}")
        
        # 禁闭能标
        Lambda_QCD = qcd.confinement_scale()
        print(f"\n禁闭能标 Λ_QCD ≈ {Lambda_QCD} GeV")
    
    @staticmethod
    def demonstrate_standard_model():
        """演示标准模型"""
        print("\n" + "=" * 70)
        print("标准模型规范结构")
        print("=" * 70)
        
        sm = StandardModelGauge()
        
        print("\n规范群: SU(3)_C × SU(2)_L × U(1)_Y")
        print(f"强耦合常数 g_s = {sm.g_s:.2f}")
        print(f"弱耦合常数 g = {sm.g:.2f}")
        print(f"超荷耦合 g' = {sm.g_prime:.2f}")
        
        # Weinberg 角
        theta_W = sm.electroweak_mixing_angle()
        print(f"\nWeinberg 角: θ_W = {np.degrees(theta_W):.2f}°")
        
        # 电磁耦合
        e = sm.electromagnetic_coupling()
        alpha_em = e**2 / (4 * np.pi)
        print(f"电磁耦合: α_EM = e²/(4π) = {alpha_em:.4f}")
        
        # Higgs 机制
        print("\nHiggs 机制 (自发对称破缺):")
        masses = sm.higgs_mechanism(246.0)
        for particle, mass in masses.items():
            print(f"  {particle}: m = {mass:.2f} GeV")
        
        # 大统一
        M_GUT = sm.unification_scale()
        print(f"\n大统一能标: M_GUT ≈ {M_GUT:.1e} GeV")


def main():
    """主函数：完整演示"""
    
    print("=" * 70)
    print("DaoMath Field Theory")
    print("从 ±0 构造场论")
    print("=" * 70)
    
    # 1. 标量场
    FieldTheoryDemo.demonstrate_scalar_field()
    
    # 2. Lagrangian 与运动方程
    FieldTheoryDemo.demonstrate_lagrangian()
    
    # 3. Noether 定理
    FieldTheoryDemo.demonstrate_noether_theorem()
    
    # 4. U(1) 规范场
    FieldTheoryDemo.demonstrate_gauge_theory()
    
    # 5. Yang-Mills 场
    FieldTheoryDemo.demonstrate_yang_mills()
    
    # 6. QCD
    FieldTheoryDemo.demonstrate_qcd()
    
    # 7. 标准模型
    FieldTheoryDemo.demonstrate_standard_model()
    
    print("\n" + "=" * 70)
    print("场论构造完成！")
    print("=" * 70)


if __name__ == "__main__":
    main()
